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If 0 le x le 2pi and sin x lt 0, which o...

If `0 le x le 2pi` and `sin x lt 0`, which of the following must be true ?
I. `cos x lt 0`
II. `csc x lt 0`
III. `|sin x + cos x|gt 0`

A

I only

B

II only

C

III only

D

I and II

Text Solution

Verified by Experts

The correct Answer is:
B
Since is negative in the third and fourth quadrants. Cosine is negative in the second and third, so I is not necessarily true. Cosecant is 1 over sine, so where sine is negative, so is cosecant. Therefore II must be true. As for III. Absolute values are always nonnegative. The issue here is where sin x + cos x can equal 0.
In fact, if x is `(7pi)/(4)`, then sin `x=-(sqrt(2))/(2)` and cos `x =(sqrt(2))/(2)`,
`sin x + cos x = 0` when `x = (7pi)/(4)`.
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